Sure, let's solve the equation step-by-step:
Given equation:
[tex]\[
\log 7 + \log (x - 4) = 1
\][/tex]
Using the properties of logarithms, specifically the product rule for logarithms, [tex]\(\log a + \log b = \log (ab)\)[/tex], we can combine the logarithms on the left-hand side:
[tex]\[
\log (7 \cdot (x - 4)) = 1
\][/tex]
This can be rewritten as:
[tex]\[
\log (7(x - 4)) = 1
\][/tex]
To solve for [tex]\(x\)[/tex], we convert the logarithmic equation to its exponential form. Recall that [tex]\(\log_b (a) = c\)[/tex] is equivalent to [tex]\(b^c = a\)[/tex]. Here, the base of the logarithm is 10 (common log [tex]\(\log\)[/tex] is base 10), so:
[tex]\[
10^1 = 7(x - 4)
\][/tex]
This simplifies to:
[tex]\[
10 = 7(x - 4)
\][/tex]
Next, we solve for [tex]\(x\)[/tex]. Start by distributing the 7:
[tex]\[
10 = 7x - 28
\][/tex]
Add 28 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
10 + 28 = 7x
\][/tex]
Simplifying the left-hand side:
[tex]\[
38 = 7x
\][/tex]
Finally, divide both sides by 7 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{38}{7}
\][/tex]
Thus, the solution to the equation [tex]\(\log 7 + \log (x - 4) = 1\)[/tex] is:
[tex]\[
x = \frac{38}{7}
\][/tex]
